Mathematical Logic Quarterly 45 (4):481-496 (1999)
Abstract |
For semi-continuous real functions we study different computability concepts defined via computability of epigraphs and hypographs. We call a real function f lower semi-computable of type one, if its open hypograph hypo is recursively enumerably open in dom × ℝ; we call f lower semi-computable of type two, if its closed epigraph Epi is recursively enumerably closed in dom × ℝ; we call f lower semi-computable of type three, if Epi is recursively closed in dom × ℝ. We show that type one and type two semi-computability are independent and that type three semi-computability plus effectively uniform continuity implies computability, which is false for type one and type two instead of type three. We show also that the integral of a type three semi-computable real function on a computable interval is not necessarily computable
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Keywords | Semi‐continuous function Computable analysis Semi‐computable function |
Categories | (categorize this paper) |
DOI | 10.1002/malq.19990450407 |
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References found in this work BETA
Theory of Recursive Functions and Effective Computability.Hartley Rogers - 1971 - Journal of Symbolic Logic 36 (1):141-146.
Effective Content of the Calculus of Variations I: Semi-Continuity and the Chattering Lemma.Xiaolin Ge & Anil Nerode - 1996 - Annals of Pure and Applied Logic 78 (1-3):127-146.
Computable Real‐Valued Functions on Recursive Open and Closed Subsets of Euclidean Space.Qing Zhou - 1996 - Mathematical Logic Quarterly 42 (1):379-409.
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